CHC-DE-VP0: Public BAO-SN-CMB Background-Stress Witness for the Homogeneous Phase-Flatness Branch

The purpose of this note is to test whether a representative homogeneous non-phantom plateau witness is excluded by one published public background posterior route. The result is deliberately reported with the label

\texttt{DE-BACKGROUND-NONEXCLUSION}.

The word ``pass'' is not used as the result label. The label means only that the declared witness remains inside the selected same-stack public posterior envelope under the stated construction.

The result does not convert the CHC-DE paper into a data-fit paper. It does not claim that CHC is preferred by DESI, Pantheon+, Planck, or ACT. It does not assign a posterior over the full CHC-DE branch family. It does not construct a sealed reconstruction stack. It does not close all possible background-data routes.

The selected public background route is a released DESI DR2 BAO cosmology chain route. DESI released the DR2 cosmology chains and supporting products for the DR2 BAO cosmology papers, enabling external reproduction and exploration of the DESI DR2 cosmology results [citation]. The selected stack includes Pantheon+ and a CMB convention containing Planck low-$\ell$TeX\ell, Planck NPIPE high-$\ell$TeX\ell CamSpec TTTEEE, and Planck--ACT DR6 lensing components as encoded in the released Cobaya route. Pantheon+ supplies the Type Ia supernova distance data surface used in the stack, with 1701 light curves of 1550 spectroscopically confirmed Type Ia supernovae [citation]. Planck PR4/NPIPE supplies the Planck reprocessed CMB background route and associated likelihood resources [citation]. ACT DR6 supplies the ACT DR6 CMB public data-product route, including maps, spectra, likelihoods, and posterior chains [citation].

The frozen public background route is center cobaya/base_w/desi-bao-all_pantheonplus_planck2018-lowl-TT-clik_planck2018-lowl-EE-clik_planck-NPIPE-highl-CamSpec-TTTEEE_planck-act-dr6-lensing center The selected public route is identified at the level of its DESI DR2 BAO chain convention, Pantheon+ component, Planck/ACT CMB convention, covariance convention, and same-stack posterior-summary surface. These source identifications are used only to define the bounded background-stress comparison and do not enlarge the manuscript claim.

Figure or table content is omitted from the web reader; use the canonical manuscript for the exact object.

Let the selected public chain provide weighted samples

\left(H_{0,i},\Omega_{{\rm m},i},w_i,r_{{\rm d},i}\right),
\qquad i=1,\ldots,N.

For each retained posterior sample, the background observables are computed on the declared grid $0\le z\le 2.5$TeX0\le z\le 2.5 using the constant-$w$TeXw flat background relation

E_i(z)=\left[\Omega_{{\rm m},i}(1+z)^3+(1-\Omega_{{\rm m},i})(1+z)^{3(1+w_i)}\right]^{1/2}.

The corresponding Hubble distance, comoving angular-diameter distance, and distance-modulus proxy are

D_{H,i}(z)=\frac{c/H_{0,i}}{E_i(z)},

D_{M,i}(z)=\int_0^z \frac{c/H_{0,i}}{E_i(z')}\,dz',

\mu_i(z)=5\log_{10}\!\left[(1+z)D_{M,i}(z)\times10^5\right].

The public-chain envelope is the pointwise posterior envelope of

E(z),\qquad D_H(z)/r_d,
\qquad D_M(z)/r_d,
\qquad \mu(z).

The selected chain summary contains 51,208 valid weighted samples. The envelope calculation uses 8,000 weighted posterior samples. The same-stack weighted parameter summaries are shown in reference.

Figure or table content is omitted from the web reader; use the canonical manuscript for the exact object.

The evaluated construction is a fixed-route public-chain comparison, not a free reconstruction. The construction has five scientific steps.

- Fix exactly one DESI DR2 base_w route and do not merge chains from different model or data conventions. - Fix the selected public posterior route, its covariance convention, and the same-stack summary convention used for the declared comparison. - Build the pointwise public-chain posterior envelope for $E(z)$TeXE(z), $D_M(z)/r_d$TeXD_M(z)/r_d, $D_H(z)/r_d$TeXD_H(z)/r_d, and $\mu(z)$TeX\mu(z) from 8,000 weighted posterior samples on the declared redshift grid. - Evaluate the homogeneous non-phantom plateau witness using the same-stack median anchors recorded in reference. - Assign only the bounded label DE-BACKGROUND-NONEXCLUSION after confirming the four observable gates and the non-phantom gate.

The route type is a declared public-chain background-stress evaluation. The same-stack envelope summary and the witness decision record are the declared support summaries for the evaluated classification; no additional classification route is invoked here.

The witness is a representative homogeneous non-phantom plateau branch with

w_{\rm eff}(z)=-1.

It is anchored to the same selected public-chain route by the median values

H_0=67.96483775,
\qquad
\Omega_{\rm m}=0.30467558,
\qquad
r_d=147.74197500000002.

The witness expansion is

E_{\rm wit}(z)=\left[\Omega_{\rm m}(1+z)^3+(1-\Omega_{\rm m})\right]^{1/2}.

The associated observables are computed from Eqs. reference, reference, and reference with the witness values inserted.

This witness is not a new fitted branch. It is a same-stack representative of the homogeneous non-phantom plateau reading admitted by the CHC-DE paper. The witness is intentionally narrower than a reconstruction of the full CHC-DE family.

For each observable, the witness is compared with the selected public-chain posterior envelope. If every witness curve lies inside the pointwise 95% envelope, the observable is assigned

\texttt{DE-BACKGROUND-NONEXCLUSION}.

If a witness leaves the 95% envelope but remains inside the wider 99.97% envelope, the observable is assigned

\texttt{DE-BACKGROUND-STRESS}.

If a witness leaves the wider envelope, it is assigned

\texttt{DE-BACKGROUND-FAIL}.

The non-phantom gate requires

w_{\rm eff}(z)\ge -1

on the declared grid. The witness has $w_{\rm eff}=-1$TeXw_{\rm eff}=-1, so the non-phantom gate is satisfied on the boundary.

The evaluated result is shown in the gate board in reference. The witness decision record reports that each observable gate lies inside both the 95% and the wider 99.97% public-chain envelopes.

Figure or table content is omitted from the web reader; use the canonical manuscript for the exact object.

The overall classification is therefore

\boxed{\DELabel{}}.

The exact reading is: a representative same-stack homogeneous non-phantom plateau witness is not excluded by the selected released public base_w posterior envelope.

The classification in reference is bounded. It does not establish a data-selected DE fit. It does not show that CHC is preferred by DESI, Pantheon+, Planck, or ACT. It does not reconstruct the full CHC-DE family. It does not assign a posterior over CHC-DE model parameters. It does not change the CHC-DE paper into a sealed reconstruction-stack paper. It does not compare all possible DESI/Pantheon+/CMB stack routes.

The result is an evaluated same-stack public background-stress witness. It remains weaker than a full-family empirical closure because only a representative same-stack homogeneous plateau witness was evaluated.

center minipage0.92 Admissible interpretation. The result supports only on the selected same-stack public BAO--SN--CMB background-stress route. Excluded interpretation. It does not alter any theorem or equation status and does not supply a data-selected CHC-DE fit, model-preference claim, sealed reconstruction stack, source-side closure, distance/clock/perturbation inference, or full CHC-DE family posterior. minipage center

The public-source summary identifies the selected data path, combined-data summary, public-chain background-stress envelope, and witness decision reported here. Large public-chain products remain external public data products and do not enlarge the scientific claim.

The compact-target branch of the root theory does not alter this background witness by notation alone. The winding invariant is carried by spatial loops, whereas the present likelihood route assumes a spatially homogeneous background. The route therefore belongs to the zero-spatial-winding sector unless a nonhomogeneous compact-field configuration is introduced and solved explicitly. In that sector the local canonical reduction used by the background equations remains exact; a nonzero winding sector would instead add a positive gradient-energy contribution bounded below by the root winding-energy theorem and could not be absorbed into the same homogeneous calibration.

The BAO, supernova, and CMB distance summaries also constitute a natural overidentification test. Once one background parameter vector is frozen, their joint prediction is a shared response map. If the observable dimension exceeds its local Jacobian rank, left-null combinations of the response must vanish to first order. Dataset-specific refitting would remove those restrictions and is therefore classified as a new model rather than a successful witness.

The closure test for the public BAO--SN--CMB background witness is applied to a dimensionless observable vector $y\in\mathbb R^m$TeXy\in\mathbb R^m formed from fixed reference scales and the declared basket of the sampled expansion, radial and transverse BAO distances, and supernova distance summaries. Let $a$TeXa range over the independent constitutive inputs comprising chain selection, posterior-envelope construction, reference cosmology, and witness coordinates.

proposition: Functional saturation, finite closure, and sector admissibility. Suppose the unrestricted prediction map $F:a\mapsto y$TeXF:a\mapsto y is continuously differentiable on a Banach space of constitutive inputs. If $D_aF$TeXD_aF is surjective and has a bounded right inverse at the calibration point, the unrestricted family is locally open in observable space and supplies no nonzero local equality restriction on $y$TeXy. Suppose instead that a single microscopic closure replaces $a$TeXa by finite parameters $\theta\in\mathbb R^p$TeX\theta\in\mathbb R^p, with profiled nuisance coordinates $\eta\in\mathbb R^q$TeX\eta\in\mathbb R^q. If

J=D_\eta F_{\rm cl}D_\theta F_{\rm cl},
 \qquad \operatorname{rank}J=r<m,

then there are $m-r$TeXm-r independent first-order restrictions

w^{\mathsf T}\delta y=0,
 \qquad w\in\ker J^{\mathsf T}.

If the rank is constant locally, these restrictions are tangent to a compatibility manifold of codimension $m-r$TeXm-r. For this sector, the finite closure is admissible only if the public chain, normalization, covariance, and witness coordinates are frozen before the transverse residuals are computed.

proof. Split surjectivity gives a bounded right inverse $R$TeXR with $D_aF\,R=I_m$TeXD_aF\,R=I_m. The Banach-space submersion theorem then makes $F$TeXF locally onto a neighborhood of the calibrated observable vector. Any smooth equality holding throughout that image must therefore vanish on an open set and contributes no model-specific local restriction. Under finite closure, the attainable first-order variations are exactly the column space of $J$TeXJ. Its orthogonal complement is $\ker J^{\mathsf T}$TeX\ker J^{\mathsf T}, whose dimension is $m-r$TeXm-r by rank--nullity, which proves reference. The constant-rank theorem supplies the stated local manifold. The sector condition is necessary because reusing a held-out summary to move the witness converts the reported nonexclusion check into calibration. Failure of that condition therefore rejects the proposed microscopic closure before parameter estimation can be counted as evidence for it.

The CHC-DE-VP0-BACKGROUND-STRESS companion note states one selected public BAO--SN--CMB background-stress route. On that route, a representative homogeneous non-phantom plateau witness remains inside the public posterior envelope for $E(z)$TeXE(z), $D_M/r_d$TeXD_M/r_d, $D_H/r_d$TeXD_H/r_d, and $\mu(z)$TeX\mu(z), and satisfies the non-phantom gate. The resulting label is . The result is a same-stack witness nonexclusion statement, not a data-selected DE fit and not a full reconstruction or closure of the CHC-DE family.

Data and code availability..

This companion manuscript uses public cosmology data products and literature-reported reference quantities as described in the text. Cited public references and companion statements, where provided, are identified by the companion source summaries cited in the text.

Funding and competing interests..

No external funding was received for this work. The author declares no competing interests.